{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T22:34:53Z","timestamp":1769726093470,"version":"3.49.0"},"reference-count":43,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2023,8,17]],"date-time":"2023-08-17T00:00:00Z","timestamp":1692230400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["SFB F65"],"award-info":[{"award-number":["SFB F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["W1245"],"award-info":[{"award-number":["W1245"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001821","name":"Vienna Science and Technology Fund","doi-asserted-by":"publisher","award":["MA16-066"],"award-info":[{"award-number":["MA16-066"]}],"id":[{"id":"10.13039\/501100001821","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,4,1]]},"abstract":"Abstract<\/jats:title>\n We present the self-consistent Pauli equation, a semi-relativistic model for charged spin-\n \n \n \n 1<\/m:mn>\n \/<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n 1\/2<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> particles with self-interaction with the electromagnetic field.\nThe Pauli equation arises as the \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n 1<\/m:mn>\n \/<\/m:mo>\n c<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n O(1\/c)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> approximation of the relativistic Dirac equation.\nThe fully relativistic self-consistent model is the Dirac\u2013Maxwell equation where the description of spin and the magnetic field arises naturally.\nIn the non-relativistic setting, the correct self-consistent equation is the Schr\u00f6dinger\u2013Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only.\nThe Schr\u00f6dinger\u2013Poisson equation also arises as the mean field limit of the \ud835\udc41-body Schr\u00f6dinger equation with Coulomb interaction.\nWe propose that the Pauli\u2013Poisson equation arises as the mean field limit \n \n \n \n N<\/m:mi>\n \u2192<\/m:mo>\n \u221e<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n N\\to\\infty<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of the linear \ud835\udc41-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation.\nWe present the semiclassical limit of the Pauli\u2013Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in \n \n \n \n 1<\/m:mn>\n \/<\/m:mo>\n c<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n 1\/c<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of the self-consistent Vlasov equation.\nThis is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb\u2013Thirring estimates.<\/jats:p>","DOI":"10.1515\/cmam-2023-0101","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T22:01:35Z","timestamp":1692223295000},"page":"453-465","source":"Crossref","is-referenced-by-count":3,"title":["Nonlinear PDE Models in Semi-relativistic Quantum Physics"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-4272-439X","authenticated-orcid":false,"given":"Jakob","family":"M\u00f6ller","sequence":"first","affiliation":[{"name":"Research Platform MMM \u201cMathematics-Magnetism-Materials\u201d c\/o Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Wien , Oskar-Morgenstern-Platz 1, 1090 Vienna , Austria"}]},{"given":"Norbert J.","family":"Mauser","sequence":"additional","affiliation":[{"name":"Research Platform MMM \u201cMathematics-Magnetism-Materials\u201d c\/o Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Wien , Oskar-Morgenstern-Platz 1, 1090 Vienna , Austria"}]}],"member":"374","published-online":{"date-parts":[[2023,8,17]]},"reference":[{"key":"2025062712372281366_j_cmam-2023-0101_ref_001","doi-asserted-by":"crossref","unstructured":"W. Bao, N. J. Mauser and H. P. Stimming,\nEffective one particle quantum dynamics of electrons: A numerical study of the Schr\u00f6dinger\u2013Poisson-\n \n \n \n X\n \u2062\n \u03b1\n \n \n \n \\mathrm{X}\\alpha\n \n model,\nCommun. Math. Sci. 1 (2003), no. 4, 809\u2013828.","DOI":"10.4310\/CMS.2003.v1.n4.a8"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_002","doi-asserted-by":"crossref","unstructured":"J.-M. Barbaroux and V. Vougalter,\nExistence and nonlinear stability of stationary states for the magnetic Schr\u00f6dinger\u2013Poisson system,\nJ. Math. Sci. (N.\u2009Y.) 219 (2016), no. 6, 874\u2013898.","DOI":"10.1007\/s10958-016-3152-z"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_003","doi-asserted-by":"crossref","unstructured":"J.-M. Barbaroux and V. Vougalter,\nOn the well-posedness of the magnetic Schr\u00f6dinger\u2013Poisson system in \n \n \n \n R\n 3\n \n \n \n \\mathbb{R}^{3}\n \n ,\nMath. Model. Nat. Phenom. 12 (2017), no. 1, 15\u201322.","DOI":"10.1051\/mmnp\/201712102"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_004","doi-asserted-by":"crossref","unstructured":"C. Bardos, N. Besse and T. T. Nguyen,\nOnsager-type conjecture and renormalized solutions for the relativistic Vlasov\u2013Maxwell system,\nQuart. Appl. Math. 78 (2020), no. 2, 193\u2013217.","DOI":"10.1090\/qam\/1549"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_005","doi-asserted-by":"crossref","unstructured":"C. Bardos, L. Erd\u0151s, F. Golse, N. Mauser and H.-T. Yau,\nDerivation of the Schr\u00f6dinger\u2013Poisson equation from the quantum \ud835\udc41-body problem,\nC. R. Math. Acad. Sci. Paris 334 (2002), no. 6, 515\u2013520.","DOI":"10.1016\/S1631-073X(02)02253-7"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_006","doi-asserted-by":"crossref","unstructured":"C. Bardos, F. Golse, A. D. Gottlieb and N. J. Mauser,\nMean field dynamics of fermions and the time-dependent Hartree\u2013Fock equation,\nJ. Math. Pures Appl. (9) 82 (2003), no. 6, 665\u2013683.","DOI":"10.1016\/S0021-7824(03)00023-0"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_007","doi-asserted-by":"crossref","unstructured":"C. Bardos, F. Golse and N. J. Mauser,\nWeak coupling limit of the \ud835\udc41-particle Schr\u00f6dinger equation,\nMethods Appl. Anal. 7 (2000), no. 2, 275\u2013293.","DOI":"10.4310\/MAA.2000.v7.n2.a2"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_008","doi-asserted-by":"crossref","unstructured":"I. Bejenaru and D. Tataru,\nGlobal wellposedness in the energy space for the Maxwell\u2013Schr\u00f6dinger system,\nComm. Math. Phys. 288 (2009), no. 1, 145\u2013198.","DOI":"10.1007\/s00220-009-0765-9"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_009","doi-asserted-by":"crossref","unstructured":"N. Benedikter, V. Jak\u0161i\u0107, M. Porta, C. Saffirio and B. Schlein,\nMean-field evolution of fermionic mixed states,\nComm. Pure Appl. 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Sci. 14 (1991), no. 1, 35\u201361.","DOI":"10.1002\/mma.1670140103"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_013","doi-asserted-by":"crossref","unstructured":"F. Castella,\n\n \n \n \n L\n 2\n \n \n \n L^{2}\n \n solutions to the Schr\u00f6dinger\u2013Poisson system: Existence, uniqueness, time behaviour, and smoothing effects,\nMath. Models Methods Appl. Sci. 7 (1997), no. 8, 1051\u20131083.","DOI":"10.1142\/S0218202597000530"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_014","doi-asserted-by":"crossref","unstructured":"L. Erd\u0151s and J. P. Solovej,\nSemiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. II. Leading order asymptotic estimates,\nComm. Math. Phys. 188 (1997), no. 3, 599\u2013656.","DOI":"10.1007\/s002200050181"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_015","doi-asserted-by":"crossref","unstructured":"L. Erd\u0151s and H.-T. 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Lubich,\nOn splitting methods for Schr\u00f6dinger\u2013Poisson and cubic nonlinear Schr\u00f6dinger equations,\nMath. Comp. 77 (2008), no. 264, 2141\u20132153.","DOI":"10.1090\/S0025-5718-08-02101-7"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_026","doi-asserted-by":"crossref","unstructured":"J. L\u00fchrmann,\nMean-field quantum dynamics with magnetic fields,\nJ. Math. Phys. 53 (2012), no. 2, Article ID 022105.","DOI":"10.1063\/1.3687024"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_027","doi-asserted-by":"crossref","unstructured":"P. A. Markowich and N. J. Mauser,\nThe classical limit of a self-consistent quantum-Vlasov equation in 3D,\nMath. Models Methods Appl. Sci. 3 (1993), no. 1, 109\u2013124.","DOI":"10.1142\/S0218202593000072"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_028","doi-asserted-by":"crossref","unstructured":"N. Masmoudi and N. J. Mauser,\nThe selfconsistent Pauli equation,\nMonatsh. Math. 132 (2001), no. 1, 19\u201324.","DOI":"10.1007\/s006050170055"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_029","doi-asserted-by":"crossref","unstructured":"N. J. Mauser,\nRigorous derivation of the Pauli equation with time-dependent electromagnetic field,\nVLSI Design 9 (1999), no. 4, 415\u2013426.","DOI":"10.1155\/1999\/89476"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_030","doi-asserted-by":"crossref","unstructured":"N. J. Mauser,\nSemi-relativistic approximations of the Dirac equation: First and second order corrections,\nTransp. Theory Stat. Phys. 29 (2000), no. 3\u20135, 449\u2013464.","DOI":"10.1080\/00411450008205884"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_031","doi-asserted-by":"crossref","unstructured":"N. J. Mauser,\nThe Schr\u00f6dinger\u2013Poisson-\n \n \n \n X\n \u2062\n \u03b1\n \n \n \n X\\alpha\n \n equation,\nAppl. Math. 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Spohn,\nKinetic equations from Hamiltonian dynamics: Markovian limits,\nRev. Modern Phys. 52 (1980), no. 3, 569\u2013615.","DOI":"10.1103\/RevModPhys.52.569"},{"key":"2025062712372281366_j_cmam-2023-0101_ref_043","unstructured":"J. Yvon,\nSur les rapports entre la th\u00e9orie des m\u00e9langes et la statistique classique,\nC. R. Acad. Sci. Paris 223 (1946), 347\u2013349."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0101\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T12:37:49Z","timestamp":1751027869000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2023-0101\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,17]]},"references-count":43,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,1,2]]},"published-print":{"date-parts":[[2024,4,1]]}},"alternative-id":["10.1515\/cmam-2023-0101"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0101","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,17]]}}}